221
*12–212.
SOLUTION
The length lof cord is
Taking the time derivative:
When AB = 50 ft,
From Eq. (1)
T
h
e g
i
r
l
at Cstan
d
s near t
h
e e
d
ge of t
h
e p
i
er an
d
pu
ll
s
i
n t
h
e
rope horizontally at a constant speed of Determine
how fast the boat approaches the pier at the instant the
rope length AB is 50 ft.
6ft>s.
A
C
x
C
x
B
6ft/s
8ft
B
8ft
B
Ans:
222
12–213.
SOLUTION
If the hydraulic cylinder Hdraws in rod BC at
determine the speed of slider A.
2ft>s,
A
BC
H
Ans:
223
12–214.
At the instant shown, the car at A is traveling at
10 m
>
s around the curve while increasing its speed at 5 m
>
s2.
The car at B is traveling at 18.5 m
>
s along the straightaway and
increasing its speed at 2 m
>
s2. Determine the relative velocity
and relative acceleration of A with respect to B at this instant.
yB fi
18.5 m
/
s
yA fi 10 m/s
100 m
45
100 m
A
B
SOLUTION
>
>
a
A
=(5 cos 45°–1 cos 45°)i+(–1 sin 45°–5 sin 45°)j
>
Ans:
vA
>
B
=13.4 m>s
>
12–215.
The motor draws in the cord at B with an acceleration of
aB = 2 m
>
s2. When sA = 1.5 m, vB = 6 m
>
s. Determine the
velocity and acceleration of the collar at this instant.
B
2 m
A
sA
SOLUTION
Position Coordinates. The position of collar A and point B are specified by
sA
and
Velocity. Taking the time derivative of Eq. (1),
ds
B
dt
=0–
1
2
(
s
A
2+4
)
–1
>
2
a
2sA
ds
A
dt b
At the instant
sA=1.5 m, vB=+6 m>s
.
vB
is positive since it is directed toward
the positive sense of
sB
.
vA=–10.0 m>s=10.0 m>sd
vA
sA
Acceleration. Taking the time derivative of Eq. (2),
dv
B
dt
=–
c
sA
a
–
1
2b
1
sA
2+4
2
– 3
>
2
a
2sA
ds
A
dt b
+
(
sA
2+4
)
– 1
>
2
ds
A
dt d
At the instant
sA=1.5 m, aB=+2 m>s2
.
aB
is positive since it is directed toward
the positive sense of
sB
. Also,
vA=–10.0 m>s
. Then
aA
sA
225
*12–216.
If
bl
oc
k
B
i
s mov
i
ng
d
own w
i
t
h
a ve
l
oc
i
ty an
d
h
as an
acceleration
determine the velocity and acceleration of
block
Ain terms of the parameters shown.
aB,
vB
SOLUTION
l=sB+3sB
2+h2
h
A
B
v
B
,a
B
s
A
Ans:
226
12–217.
The crate Cis being lifted by moving the roller at A
downward with a constant speed of along the
guide. Determine the velocity and acceleration of the crate
at the instant When the roller is at B, the crate
rests on the ground. Neglect the size of the pulley in the
calculation. Hint: Relate the coordinates and using
the problem geometry, then take the first and second time
derivatives.
xA
xC
s=1m.
vA=2m>s
SOLUTION
, and when
Thus,
l=8m s=1m
xC+2xA
2+(4)2=l
s
C
A
4m
4m
B
x
A
x
C
,
Ans:
227
12–218.
Two planes, A and B, are flying at the same altitude. If their
velocities are
vA=500 km>h
and
vB=700 km>h
such
that the angle between their straight-line courses is u
=60°,
determine the velocity of plane B with respect to plane A.A
B
vA fi 500 km/h
vB fi 700 km/h
60
SOLUTION
Relative Velocity. Express vA and vB in Cartesian vector form,
Applying the relative velocity equation.
Ans:
228
12–219.
SOLUTION
(aB)n=v2
A
r=402
0.5 =3200 mi>h2(aB)t=1200 mi>h2
vB= –40 cos 30°i+40 sin 30°j={–34.64i+20j}mi>h
of 0.5 mi.
vA=55 mi/h
A30°
Ans:
229
*12–220.
The boat can travel with a speed of 16 km
>
h in still water. The
point of destination is located along the dashed line. If the
water is moving at 4 km
>
h, determine the bearing angle
u
at
which the boat must travel to stay on course.
vW 4 km/h
70
u
SOLUTION
v
=v
+v
Ans:
230
12–221.
SOLUTION
Relative Ve locity:
Thus, the magnitude of the relative velocity is
And its direction is
vA>B
Two boats leave the pier Pat the same time and travel in
the directions shown. If and ,
determine the velocity of boat Arelative to boat B. How
long after leaving the pier will the boats be 1500 ft apart?
vB=30 ft>svA=40 ft>s
y
x
B
A
v
B
= 30 ft/s
v
A
= 40 ft/s
45°
30°
Ans:
231
12–222.
SOLUTION
Solution I
Vector Analysis:For the first case, the velocity of the car and the velocity of the wind
relative to the car expressed in Cartesian vector form are and
. Applying the relative velocity equation, we have
Equating Eqs. (1) and (2) and then the iand jcomponents,
Substituting the result of into Eq. (1),
(vw>c)1
vW>C=(vW>C)1i
vc=[50j]km>h
A car is traveling north along a straight road at 50 km>h. An
instrument in the car indicates that the wind is coming from
the east. If the car’s speed is 80 km>h, the instrument
indicates that the wind is coming from the northeast. Deter-
mine the speed and direction of the wind.
Ans:
232
12–223.
Two boats leave the shore at the same time and travel in the
directions shown. If
vA=10 m>s
and
vB=15 m>s,
determine the velocity of boat A with respect to boat B. How
long after leaving the shore will the boats be 600 m apart?
SOLUTION
Relative Velocity. The velocity triangle shown in Fig. a is drawn based on the relative
v
Then, the sine law gives
The direction of
v
A
>
B is defined by
Applying the relative velocity equation
v
v
Thus the magnitude of
v
A
>
B is
A
O
B
vA fi 10 m/s
vB fi 15 m/s
30
45
Ans: